Let \(N\)  be a set of vertices. A mapping \(\mathcal {F}\)  is a collection of local multiscores for \(N\) if \(\mathcal {F}(v) \subseteq 2^{N \setminus \lbrace v\rbrace } \times {N}_0 \times {N}_0\)  for each \(v\in N\) .
Intuitively, if \((P,s,\omega )\in \mathcal {F}(v)\)  for some vertex \(v \in N\) , then choosing \(P\)  as the parent set of \(v\)  may simultaneously give a local score of \(s\)  and a local weight of \(\omega \) .
For the same parent set \(P\) , there might be another local multiscore \((P,s^{\prime },\omega ^{\prime })\in \mathcal {F}(v)\)  with a different local score and a different local weight.
Throughout this work we assume that for every vertex \(v\)  there exists some \(s\in {N}_0\)  such that \((\emptyset , s,0) \in \mathcal {F}(v)\) , that is, every vertex has a local multiscore for the empty parent set with weight zero.
Given \(v \in N\)  and \(\mathcal {F}\) , the possible parent sets are defined by \(\mathcal {P}_\mathcal {F}(v):=\lbrace P \mid \exists s\in {N}_0:\exists \omega \in {N}_0: (P,s,\omega ) \in \mathcal {F}(v)\rbrace \) . Given \(N\)  and \(\mathcal {F}\) , the directed graph \(S_\mathcal {F}:= (N,A_\mathcal {F})\)  with \(A_\mathcal {F}:=\lbrace (u,v) \mid \exists P \in \mathcal {P}_\mathcal {F}(v): u \in P\rbrace \)  is called the superstructure of \(N\)  and \(\mathcal {F}\)  [1]}.
